CHAPTER
5
You have seen different types of functions and how these functions can mathematically model the real world. Many sinusoidal and periodic patterns occur within nature. Movement on the surface of Earth, such as earthquakes, and stresses within Earth can cause rocks to fold into a sinusoidal pattern. Geologists and structural engineers study models of trigonometric functions to help them understand these formations. In this chapter, you will study trigonometric functions for which the function values repeat at regular intervals.
Trigonometric
Functions and
Graphs
Key Terms periodic function period
sinusoidal curve amplitude
vertical displacement phase shift
Career Link
A geologist studies the composition, structure, and history of Earth’s surface to determine the processes affecting the development of Earth. Geologists apply their knowledge of physics, chemistry, biology, and mathematics to explain these phenomena. Geological engineers apply geological knowledge to projects such as dam, tunnel, and building construction.
To learn more about a career as a geologist, go to
www.mcgrawhill.ca/school/learningcentres and follow the links.
earn more ab We b Link
5.1
Graphing Sine and
Cosine Functions
Focus on . . .
sketching the graphs of
• y = sin x and y = cos x determining the characteristics of the graphs •
of y = sin x and y = cos x
demonstrating an understanding of the effects •
of vertical and horizontal stretches on the graphs of sinusoidal functions
solving a problem by analysing the graph of a •
trigonometric function
Many natural phenomena are cyclic, such as the tides of the ocean, the orbit of Earth around the Sun, and the growth and decline in animal populations. What other examples of cyclic natural phenomena can you describe?
You can model these types of natural behaviour with periodic functions such as sine and cosine functions.
1. a) Copy and complete the table. Use your knowledge of special angles to determine exact values for each trigonometric ratio. Then, determine the approximate values, to two decimal places. One row has been completed for you.
Angle, θ y = sin θ y = cos θ
0
π_6 _12 = 0.50 √ __ 3 _
2 ≈ 0.87
π_4
π_
3
π_2
b) Extend the table to include multiples of the special angles in the other three quadrants.
Investigate the Sine and Cosine Functions
Materials
grid paper •
ruler •
The Bay of Fundy, between New Brunswick and Nova Scotia, has the highest tides in the world. The highest recorded tidal range is 17 m at Burntcoat Head, Nova Scotia.
Did You Know?
The Hopewell Rocks on the Bay of Fundy coastline are sculpted by the cyclic tides. ch as the tides of the
2. a) Graph y= sin θ on the interval θ∈ [0, 2π]
b) Summarize the following characteristics of the function y= sin θ. • the maximum value and the minimum value
• the interval over which the pattern of the function repeats • the zeros of the function in the interval θ∈ [0, 2π]
• the y-intercept • the domain and range
3. Graph y= cos θ on the interval θ∈ [0, 2π] and create a summary similar to the one you developed in step 2b).
Reflect and Respond
4. a) Suppose that you extended the graph of y= sin θ to the right of 2π. Predict the shape of the graph. Use a calculator to investigate a few points to the right of 2π. At what value of θ will the next cycle end?
b) Suppose that you extended the graph of y= sin θ to the left of 0. Predict the shape of the graph. Use a calculator to investigate a few points to the left of 0. At what value of θ will the next cycle end?
5. Repeat step 4 for y= cos θ.
Sine and cosine functions are periodic functions. The values of these functions repeat over a specified period.
A sine graph is a graph of the function y= sin θ. You can also describe a sine graph as a sinusoidal curve.
y
π
-π 2π
-2π
0.5
-0.5
-1 1
0 π_
2 3π__2 5π__2 -π2_
Period
Period One Cycle
3π __ 2 -5π __ 2
-y = sin θ
θ
Trigonometric functions are sometimes called circular because they are based on the unit circle.
Link the Ideas
periodic function a function that • repeats
itself over regular intervals (cycles) of its domain
period
the length of the •
interval of the domain over which a graph repeats itself
the horizontal length of •
one cycle on a periodic graph
sinusoidal curve the name given to a •
curve that fluctuates back and forth like a sine graph
a curve that oscillates •
repeatedly up and down from a centre line The sine function is based upon one of the trigonometric ratios originally calculated by the astronomer Hipparchus of Nicaea in the second century B.C.E. He was trying to make sense of the movement of the stars and the moon in the night sky.
Did You Know?
The sine function, y= sin θ, relates the measure of angle θ in standard position to the y-coordinate of the point P where the terminal arm of the angle intersects the unit circle.
2π
0
-1 1 y y
θ π_
2 π
π 0, 2π 3__π
2
7π
__ 4 3π
__ 2 3__π
4
5π
__ 4
π_ 2
π_ 4
y = sin θ P
The cosine function, y= cos θ, relates the measure of angle θ in standard position to the x-coordinate of the point P where the terminal arm of the angle intersects the unit circle.
y
0 x
P
P
2π y
x π
-1 1
0 π_
2 3π__2
π
_ 3, cosπ_3
(
)
7π__
6, cos7π__6
(
)
The coordinates of point P repeat after point P travels completely around the unit circle. The unit circle has a circumference of 2π. Therefore, the smallest distance before the cycle of values for the functions y= sin θ or y= cos θ begins to repeat is 2π. This distance is the period of sin θ and cos θ.
Graph a Periodic Function
Sketch the graph of y= sin θ for 0° ≤θ≤ 360° or 0 ≤θ≤ 2π. Describe its characteristics.
Solution
To sketch the graph of the sine function for 0° ≤θ≤ 360° or 0 ≤θ≤ 2π, select values of θ and determine the corresponding values of sin θ. Plot the points and join them with a smooth curve.
θ
Degrees 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360°
Radians 0 _ π6 _ π4 _ π3 _ π2 2_π
3 3π _ 4 5π _ 6 π 7π _ 6 5π _ 4 4π _ 3 3π _ 2 5π _ 3 7π _ 4 11π _
6 2π
sin θ 0 _1
2
√ __2
_ 2 √ __ 3 _ 2 1 √ __ 3 _ 2
√ __2
_
2 1
_
2 0 - 1 _ 2 -
√ __2
_
2 -
√ __3
_
2 -1 -
√ __3
_
2 -
√ __2
_
2 - _ 1 2 0
y
θ 60° 90° 120° 150°
30° 1
-1
0 180° 210° 240° 270° 300° 330° 360°
y = sin θ
y
θ 1
-1 0
y = sin θ
2π π
π_
6 π_3 π2_ 2π__3 5π__6 7π__6 4π__3 3π__2 5π__3 11π___6
From the graph of the sine function, you can make general observations about the characteristics of the sine curve:
The curve is periodic. •
• The curve is continuous. The domain is {
• θ | θ ∈ R}.
The range is {
• y| -1 ≤y≤ 1, y∈ R}.
The maximum value is
• +1.
The minimum value is
• -1.
The
• amplitudeof the curve is 1. The period is 360° or 2
• π.
The
• y-intercept is 0.
• In degrees, the θ-intercepts are
…, -540°, -360°, -180°, 0°, 180°, 360°, …, or 180°n, where n∈ I.
The θ-intercepts, in radians, are …, -3π, -2π, -π, 0, π, 2π, …, or nπ, where n∈ I.
Your Turn
Sketch the graph of y= cos θ for 0° ≤ θ ≤ 360°. Describe its characteristics.
The Indo-Asian mathematician Aryabhata (476—550) made tables of half-chords that are now known as sine and cosine tables.
Did You Know?
Which points would you determine to be the key points for sketching a graph of the sine function?
amplitude (of a sinusoidal function)
the maximum vertical •
distance the graph of a sinusoidal function varies above and below the horizontal central axis of the curve Look for a
pattern in the values.
Determine the Amplitude of a Sine Function
Any function of the form y=af(x) is related to y=f(x) by a vertical stretch of a factor |a| about the x-axis, including the sine and cosine functions. If a< 0, the function is also reflected in the x-axis.
a) On the same set of axes, graph y= 3 sin x, y= 0.5 sin x,and y=-2 sin x for 0 ≤x≤ 2π.
b) State the amplitude for each function.
c) Compare each graph to the graph of y= sin x. Consider the period, amplitude, domain, and range.
Solution
a) Method 1: Graph Using Transformations Sketch the graph of y= sin x.
For the graph of y= 3 sin x, apply a vertical stretch by a factor of 3. For the graph of y= 0.5 sin x, apply a vertical stretch by a factor of 0.5. For the graph of y=-2 sin x, reflect in the x-axis and apply a vertical stretch by a factor of 2.
y
x
2π π
-2
-3 -1 1 2 3
0 π_
4 π_2 3π__4 5π__4 3π__2 7π__4
y= 3 sin x
y = sin x
y = 0.5 sin x
y = -2 sin x
Method 2: Use a Graphing Calculator Select radian mode.
Example 2
Use the following window settings:
x:
[
0, 2π, _π4]
b) Determine the amplitude of a sine function using the formula Amplitude = maximum value _______ - minimum value
2 .
The amplitude of y= sin x is 1__ - (-1) 2 , or 1.
The amplitude of y= 3 sin x is 3__ - (-3) 2 , or 3.
The amplitude of y= 0.5 sin x is 0.5___ - (-0.5)
2 , or 0.5.
The amplitude of y=-2 sin x is 2__ - (-2) 2 , or 2.
c) Function Period Amplitude Specified Domain Range
y = sin x 2π 1 {x | 0 ≤ x ≤ 2π, x ∈ R} {y |-1 ≤ y ≤ 1, y ∈ R} y = 3 sin x 2π 3 {x | 0 ≤ x ≤ 2π, x ∈ R} {y |-3 ≤ y ≤ 3, y ∈ R} y = 0.5 sin x 2π 0.5 {x | 0 ≤ x ≤ 2π, x ∈ R} {y |-0.5 ≤ y ≤ 0.5, y ∈ R} y =-2 sin x 2π 2 {x | 0 ≤ x ≤ 2π, x ∈ R} {y |-2 ≤ y ≤ 2, y ∈ R} Changing the value of a affects the amplitude of a sinusoidal function. For the function y=a sin x, the amplitude is |a|.
Your Turn
a) On the same set of axes, graph y= 6 cos x and y=-4 cos x for 0 ≤x≤ 2π.
b) State the amplitude for each graph.
c) Compare your graphs to the graph of y= cos x. Consider the period, amplitude, domain, and range.
d) What is the amplitude of the function y= 1.5 cos x?
Period of y = sin bx or y = cos bx
The graph of a function of the form y= sin bx or y= cos bx for b≠ 0 has a period different from 2π when |b|≠ 1. To show this, remember that sin bx or cos bx will take on all possible values as bx ranges from 0 to 2π. Therefore, to determine the period of either of these functions, solve the compound inequality as follows.
0 ≤x≤ 2π
0 ≤ |b|x ≤ 2π
0 ≤x≤ 2_ π
|b|
Solving this inequality determines the length of a cycle for the sinusoidal
curve, where the start of a cycle of y= sin bx is 0 and the end is 2_ π
|b| .
Determine the period, or length of the cycle, by finding the distance from 0 to 2_ π
|b| . Thus, the period for y= sin bx or y= cos bx is 2 π
_
|b| ,
in radians, or 360° _
|b| , in degrees.
How is the amplitude related to the range of the function?
Begin with the interval of one cycle of y = sin x or y = cos x.
Replace x with |b|x for the interval of one cycle of y = sin bx or y = cos bx.
Divide by |b|.
Why do you use |b| to determine the period?
Determine the Period of a Sine Function
Any function of the form y=f(bx) is related to y=f(x) by a horizontal stretch by a factor of 1 _
|b| about the y-axis, including
the sine and cosine functions. If b< 0, then the function is also reflected in the y-axis.
a) Sketch the graph of the function y= sin 4x for 0 ≤x≤ 360°. State the period of the function and compare the graph to the graph of y= sin x.
b) Sketch the graph of the function y= sin 1 _
2 x for 0 ≤x≤ 4π. State the period of the function and compare the graph to the graph of y= sin x.
Solution
a) Sketch the graph of y= sin x.
For the graph of y= sin 4x, apply a horizontal stretch by a factor of 1 _ 4 .
y
x
-1
-2 1 2
0 90° 180° 270° 360°
y= sin 4x
y = sin x
From the graph of y= sin 4x, the period is 90°.
You can also determine this using the formula Period = 360° _
|b| .
Period = 360° _
|b|
Period = 360_ °
|4|
Period = 360° _ 4 Period = 90°
Compared to the graph of y= sin x, the graph of y= sin 4x has the same amplitude, domain, and range, but a different period.
Example 3
To find the period of a function, start from any point on the graph (for example, the y-intercept) and determine the length of the interval until one cycle is complete.
b) Sketch the graph of y= sin x. For the graph of y= sin 1 _
2 x, apply a horizontal stretch by a factor of 2. y
x
2π 3π 4π
π -1
1
0 π_
2 3π__2 5π__2 7π__2 y = sin x
y = sin x1_2
From the graph, the period for y= sin 1 _ 2 x is 4π. Using the formula,
Period = 2_ π
|b|
Period = 2_ π
|
_ 12
|
Period = 2
_
π 1 _ 2Period = 4π
Compared to the graph of y= sin x, the graph of y= sin 1 _
2 x has the same amplitude, domain, and range, but a different period.
Changing the value of b affects the period of a sinusoidal function.
Your Turn
a) Sketch the graph of the function y= cos 3x for 0 ≤x≤ 360°. State the period of the function and compare the graph to the graph of y= cos x.
b) Sketch the graph of the function y= cos 1 _
3 x for 0 ≤x≤ 6π. State the period of the function and compare the graph to the graph of y= cos x.
c) What is the period of the graph of y= cos (-3x)?
Sketch the Graph of y = a cos bx
a) Sketch the graph of y=-3 cos 2x for at least one cycle.
b) Determine • the amplitude • the period
• the maximum and minimum values • the x-intercepts and the y-intercept • the domain and range
Substitute 1_2 for b.
Example 4
Solution
a) Method 1: Graph Using Transformations
Compared to the graph of y= cos x, the graph of y=-3 cos 2x is stretched horizontally by a factor of 1 _
2 about the y-axis, stretched vertically by a factor of 3 about the x-axis, and reflected in the x-axis.
Begin with the graph of y= cos x. Apply a horizontal stretch of 1 _
2 about the y-axis.
3π 4π
2π y
x π
-2
-3 -1 1 2 3
0
y = cos 2x
y = cos x π_
2 3π__2 5π__2 7π__2
Then, apply a vertical stretch by a factor of 3.
3π 4π
2π y
x π
-2
-3 -1 1 2 3
0
y = cos 2x
π_
2 3π__2 5π__2 7π__2
y = 3 cos 2x
Finally, reflect the graph of y= 3 cos 2x in the x-axis.
3π 4π
2π y
x π
-2
-3 -1 1 2 3
0
y= -3 cos 2x
π_
2 3π__2 5π__2 7π__2
y = 3 cos 2x
Why is the horizontal stretch
Method 2: Graph Using Key Points
This method is based on the fact that one cycle of a cosine function
y= cos bx, from 0 to 2_ π
|b| , includes two x-intercepts, two maximums,
and a minimum. These five points divide the period into quarters.
Compare y=-3 cos 2x to y=a cos bx.
Since a=-3, the amplitude is |-3|, or 3. Thus, the maximum value is 3 and the minimum value is -3.
Since b= 2, the period is 2_ π
|2| , or π. One cycle will start at x= 0 and
end at x=π. Divide this cycle into four equal segments using the values 0, _ π
4 ,
π
_
2 , 3
π
_
4 , and π for x. The key points are (0, -3),
(
π_4 , 0
)
,(
_ π2 , 3
)
,(
3π
_
4 , 0
)
, and (π, -3).Connect the points in a smooth curve and sketch the graph through one cycle. The graph of y=-3 cos 2x repeats every π units in either direction.
3π 4π
2π y
x π
-2
-3 -1 1 2 3
0
y = -3 cos 2x
π_
2 3π__2 5π__2 7π__2
b) The amplitude of y=-3 cos 2x is 3. The period is π.
The maximum value is 3. The minimum value is -3 The y-intercept is -3. The x-intercepts are _ π
4 , 3
π
_
4 , 5
π
_
4 , 7
π
_
4 or
π
_
4 +
π
_
2 n, n∈ I. The domain of the function is {x|x∈ R}.
The range of the function is {y|-3 ≤y≤ 3, y∈ R}.
How do you know where the maximums or minimums will occur?
Why are there two minimums instead of two maximums?
Your Turn
a) Graph y= 3 sin 4x, showing at least two cycles.
b) Determine • the amplitude • the period
• the maximum and minimum values • the x-intercepts and the y-intercept • the domain and range
Key Ideas
To sketch the graphs of y= sin θ and y= cos θfor 0° ≤θ≤ 360° or 0 ≤θ≤ 2π, determine the coordinates of the key points representing the θ-intercepts, maximum(s), and minimum(s).
y
θ 2π π
-1 1
0 π_
2 3π__2 y = sin θ
The maximum value is +1. The minimum value is -1. The amplitude is 1. The period is 2π. The y-intercept is 0.
The θ-intercepts for the cycle shown are 0, π, and 2π. The domain of y = sin θ is {θ|θ∈ R}.
The range of y = sin θ is {y |-1 ≤ y ≤ 1, y ∈ R}.
y
θ 2π π
-1 1
0 π_
2 3π__2 y = cos θ
How are the characteristics different for y = cos θ?
Determine the amplitude and period of a sinusoidal function of the form y=a sin bx or y=a cos bx by inspecting graphs or directly from the sinusoidal function.
You can determine the amplitude using the formula
Amplitude = maximum value _______ - minimum value
2 .
The amplitude is given by |a|.
You can change the amplitude of a function by varying the value of a.
The period is the horizontal length of one cycle on the graph of a function. It is given by 2_ π
|b| or 360° _ |b| .
You can change the period of a function by varying the value of b.
How can you determine the amplitude from the graph of the sine function? cosine function?
How can you identify the period on the graph of a sine function? cosine function?
Check Your Understanding
Practise
1. a) State the five key points for y= sin x that occur in one complete cycle from 0 to 2π.
b) Use the key points to sketch the graph of y= sin x for -2π≤x≤ 2π. Indicate the key points on your graph.
c) What are the x-intercepts of the graph?
d) What is the y-intercept of the graph?
e) What is the maximum value of the graph? the minimum value?
2. a) State the five key points for y= cos x that occur in one complete cycle from 0 to 2π.
b) Use the key points to sketch a graph of y= cos x for -2π≤x≤ 2π. Indicate the key points on your graph.
c) What are the x-intercepts of the graph?
d) What is the y-intercept of the graph?
e) What is the maximum value of the graph? the minimum value?
3. Copy and complete the table of properties for y= sin x and y= cos x for all real numbers.
Property y = sin x y = cos x
maximum
minimum
amplitude
period
domain
range
y-intercept
x-intercepts
4. State the amplitude of each periodic function. Sketch the graph of each function.
a) y= 2 sin θ b) y= 1 _ 2 cos θ
c) y=- _ 1
3 sin x d) y=-6 cos x
5. State the period for each periodic function, in degrees and in radians. Sketch the graph of each function.
a) y= sin 4θ b) y= cos 1 _ 3 θ
c) y= sin 2 _
3 x d) y= cos 6x
Apply
6. Match each function with its graph.
a) y= 3 cos x b) y= cos 3x c) y=-sin x d) y=-cos x
A y
2π π
-2 2
0 π_
2 3π__2 x
B y
2π π
-1 1
0 π_
2 3π__2 x
C y
2π π
-1 1
0 π_
2 3__2π x
D y
x π
-1 1
0 π_
3 2__3π 4__3π
7. Determine the amplitude of each function. Then, use the language of transformations to describe how each graph is related to the graph of y= sin x.
a) y= 3 sin x b) y=-5 sin x c) y= 0.15 sin x d) y=- _ 2
3 sin x
8. Determine the period (in degrees) of each function. Then, use the language of transformations to describe how each graph is related to the graph of y= cos x.
a) y= cos 2x b) y= cos (-3x)
c) y= cos 1 _
4 x d) y= cos 2 _ 3 x
9. Without graphing, determine the amplitude and period of each function. State the period in degrees and in radians.
a) y = 2 sin x b) y =-4 cos 2x c) y= 5 _
3 sin
(
- _ 2 3 x)
d) y= 3 cos 1 _ 2 x10. a) Determine the period and the amplitude of each function in the graph.
y
x π 2π 3π
-1
-2 1 2
0 π_
2 3__2π 5__2π 7__2π 4π
A
B
b) Write an equation in the form y=a sin bx or y=a cos bx for each function.
c) Explain your choice of either sine or cosine for each function.
11. Sketch the graph of each function over the interval [-360°, 360°]. For each function, clearly label the maximum and minimum values, the x-intercepts, the y-intercept, the period, and the range.
a) y= 2 cos x b) y=-3 sin x c) y= 1 _
2 sin x d) y=- _ 3 4 cos x
12. The points indicated on the graph shown represent the x-intercepts and the maximum and minimum values.
x
A B
C
D E
F
a) Determine the coordinates of points B, C, D, and E if y= 3 sin 2x and A has coordinates (0, 0).
b) Determine the coordinates of points C, D, E, and F if y= 2 cos x and B has coordinates (0, 2).
c) Determine the coordinates of points B, C, D, and E if y= sin 1 _
2 x and A has coordinates (-4π, 0).
13. The second harmonic in sound is given by f(x) = sin 2x, while the third harmonic is given by f(x) = sin 3x. Sketch the curves and compare the graphs of the second and third harmonics for -2π≤x≤ 2π.
A harmonic is a wave whose frequency is an integral multiple of the fundamental frequency. The fundamental frequency of a periodic wave is the inverse of the period length.
Did You Know?
14. Sounds heard by the human ear are vibrations created by different air pressures. Musical sounds are regular or periodic vibrations. Pure tones will produce single sine waves on an oscilloscope. Determine the amplitude and period of each single sine wave shown.
a) y
x
π 2π
-4
-2 2 4
0 π_ 3
b) y x π 2π -4 -2 2 4
0 π_ 3
-π_3 2π__3 4π__3 5π__3
Pure tone audiometry is a hearing test used to measure the hearing threshold levels of a patient. This test determines if there is hearing loss. Pure tone audiometry relies on a patient’s response to pure tone stimuli.
Did You Know?
15. Systolic and diastolic pressures mark the upper and lower limits in the changes in blood pressure that produce a pulse. The length of time between the peaks relates to the period of the pulse.
0.8 1.6 Systolic Pressure Diastolic Pressure 2.4 3.2 40 80 120 0 Pressure
(in millimetres of mercury)
Time (in seconds)
Blood Pressure Variation 160
a) Determine the period and amplitude of the graph.
b) Determine the pulse rate (number of beats per minute) for this person.
16. MINI LAB Follow these steps
to draw a sine curve. Step 1 Draw a large circle.
a) Mark the centre of the circle.
b) Use a protractor and mark every 15° from 0° to 180° along the circumference of the circle.
c) Draw a line radiating from the centre of the circle to each mark.
d) Draw a vertical line to complete a right triangle for each of the angles that you measured.
Step 2 Recall that the sine ratio is the length of the opposite side divided by the length of the hypotenuse. The hypotenuse of each triangle is the radius of the circle. Measure the length of the opposite side for each triangle and complete a table similar to the one shown.
Angle,
x Opposite Hypotenuse sin x =
opposite __ hypotenuse 0° 15° 30° 45°
Step 3 Draw a coordinate grid on a sheet of grid paper.
a) Label the x-axis from 0° to 360° in increments of 15°.
b) Label the y-axis from -1 to +1.
c) Create a scatter plot of points from your table. Join the dots with a smooth curve.
Step 4 Use one of the following methods to complete one cycle of the sine graph:
• complete the diagram from 180° to 360°
• extend the table by measuring the lengths of the sides of the triangle
• use the symmetry of the sine curve to complete the cycle
Materials paper • protractor • compass • ruler • grid paper •
17. Sketch one cycle of a sinusoidal curve with the given amplitude and period and passing through the given point.
a) amplitude 2, period 180°, point (0, 0)
b) amplitude 1.5, period 540°, point (0, 0)
18. The graphs of y= sin θ and y= cos θ show the coordinates of one point. Determine the coordinates of four other points on the graph with the same y-coordinate as the point shown. Explain how you determined the θ-coordinates.
a) y
π
-π 2π
-1 1 2
0 π_ 2
-π2_ 3__2π -3__2π
3__π
4, __22
(
)
θ
b) y
π
-π 2π
-1 2
1
0 π_ 2
-π2_ 3__2π -3__2π
,
π_ 6 __23
( )
θ
19. Graph y= sin θ and y= cos θon the same set of axes for -2π≤θ≤ 2π.
a) How are the two graphs similar?
b) How are they different?
c) What transformation could you apply to make them the same graph?
Extend
20. If y =f(x) has a period of 6, determine the period of y=f
(
1 _2 x
)
.21. Determine the period, in radians, of each function using two different methods.
a) y=-2 sin 3x b) y=- _ 2
3 cos
π
_
6 x
22. If sin θ= 0.3, determine the value of sin θ+ sin (θ+ 2π) + sin (θ+ 4π).
23. Consider the function y= √ _____sin x .
a) Use the graph of y= sin x to sketch a prediction for the shape of the graph of y= √ _____sin x .
b) Use graphing technology or grid paper and a table of values to check your prediction. Resolve any differences.
c) How do you think the graph of y= √ _________sin x+ 1 will differ from the graph of y= √ _____sin x ?
d) Graph y= √ _________sin x+ 1 and compare it to your prediction.
24. Is the function f(x) = 5 cos x+ 3 sin x sinusoidal? If it is sinusoidal, state the period of the function.
In 1822, French mathematician Joseph Fourier discovered that any wave could be modelled as a combination of different types of sine waves. This model applies even to unusual waves such as square waves and highly irregular waves such as human speech. The discipline of reducing a complex wave to a combination of sine waves is called Fourier analysis and is fundamental to many of the sciences.
Did You Know?
C1 MINI LAB Explore the relationship between the unit circle and the sine and cosine graphs with a graphing calculator.
Step 1 In the first list, enter the angle values from 0 to 2π by increments of _ π
12 . In the second and third lists, calculate the cosine and sine of the angles in the first list, respectively.
Step 2 Graph the second and third lists for the unit circle.
Step 3 Graph the first and third lists for the sine curve.
Step 4 Graph the first and second lists for the cosine curve.
Step 5 a) Use the trace feature on the graphing calculator and trace around the unit circle. What do you notice about the points that you trace? What do they represent?
b) Move the cursor to trace the sine or cosine curve. How do the points on the graph of the sine or cosine curve relate to the points on the unit circle? Explain.
C2 The value of (cos θ)2+ (sin θ)2 appears to be constant no matter the value of θ. What is the value of the constant? Why is the value constant? (Hint: Use the unit circle and the Pythagorean theorem in your explanation.)
C3 The graph of y=f(x) is sinusoidal with a period of 40° passing through the point (4, 0). Decide whether each of the following can be determined from this information, and justify your answer.
a) f(0)
b) f(4)
c) f(84)
C4 Identify the regions that each of the following characteristics fall into.
Sine Cosine y= sin x y= cos x
Sine and Cosine
a) domain {x|x∈ R}
b) range {y|-1 ≤y≤ 1, y∈ R}
c) period is 2π
d) amplitude is 1
e) x-intercepts are n(180°), n∈ I
f) x-intercepts are 90° +n(180°), n∈ I
g) y-intercept is 1
h) y-intercept is 0
i) passes through point (0, 1)
j) passes through point (0, 0)
k) a maximum value occurs at (360°, 1)
l) a maximum value occurs at (90°, 1)
m) y
x 0
n)
x y
0
C5 a) Sketch the graph of y=|cos x| for
-2π≤x≤ 2π. How does the graph compare to the graph of y= cos x?
b) Sketch the graph of y=|sin x| for
-2π≤x≤ 2π. How does the graph compare to the graph of y= sin x?
A: Graph y = sin θ+ d or y = cos θ+ d
1. On the same set of axes, sketch the graphs of the following functions for 0° ≤θ≤ 360°.
y= sin θ
y= sin θ+ 1 y= sin θ- 2
2. Using the language of transformations, compare the graphs of y= sin θ+ 1 and y= sin θ− 2 to the graph of y= sin θ.
3. Predict what the graphs of y= sin θ+ 3 and y= sin θ- 4 will look like. Justify your predictions.
Investigate Transformations of Sinusoidal Functions
Materials
grid paper •
graphing technology •
Transformations of
Sinusoidal Functions
Focus on . . .
graphing and transforming sinusoidal functions •
identifying the domain, range, phase shift, period, •
amplitude, and vertical displacement of sinusoidal functions
developing equations of sinusoidal functions, •
expressed in radian and degree measure, from graphs and descriptions
solving problems graphically that can be modelled using •
sinusoidal functions
recognizing that more than one equation can be used to •
represent the graph of a sinusoidal function
The motion of a body attached to a
suspended spring, the motion of the plucked string of a musical instrument, and the pendulum of a clock produce oscillatory motion that you can model with sinusoidal functions. To use the functions y= sin x and y= cos x in applied situations, such as these and the ones in the images shown, you need to be able to transform the functions.
The pistons and connecting rods of a steam train drive the wheels with a motion that is sinusoidal. Electric power and the
light waves it generates are sinusoidal waveforms.
Ocean waves created by the winds may be modelled by sinusoidal curves.
Reflect and Respond
4. a) What effect does the parameter d in the function y= sin θ+d have on the graph of y= sin θ when d> 0?
b) What effect does the parameter d in the function y= sin θ+d have on the graph of y= sin θ when d< 0?
5. a) Predict the effect varying the parameter d in the function y= cos θ+d has on the graph of y= cos θ.
b) Use a graph to verify your prediction.
B: Graph y = cos (θ- c) or y = sin (θ- c) Using Technology
6. On the same set of axes, sketch the graphs of the following functions for -π≤θ≤ 2π.
y= cos θ
y= cos
(
θ+ _ π 2)
y= cos (θ-π)7. Using the language of transformations, compare the graphs of y= cos
(
θ+ _ π2
)
and y= cos (θ-π)to the graph of y= cos θ.8. Predict what the graphs of y= cos
(
θ- _ π2
)
and y= cos(
θ+ 3π
_
2
)
will look like. Justify your predictions.Reflect and Respond
9. a) What effect does the parameter c in the function y= cos (θ-c) have on the graph of y= cos θ when c> 0?
b) What effect does the parameter c in the function y= cos (θ-c) have on the graph of y= cos θ when c< 0?
10. a) Predict the effect varying the parameter c in the function y= sin (θ-c) has on the graph of y= sin θ.
b) Use a graph to verify your prediction.
You can translate graphs of functions up or down or left or right and stretch them vertically and/or horizontally. The rules that you have applied to the transformations of functions also apply to transformations of sinusoidal curves.
Link the Ideas
Graph y = sin (x - c) + d
a) Sketch the graph of the function y= sin (x- 30°) + 3.
b) What are the domain and range of the function?
c) Use the language of transformations to compare your graph to the graph of y= sin x.
Solution
a)
360° 300° 3
4 y
60° 120° 180° 240° -1
1 2
0 x
b) Domain: {x|x∈ R}
Range: {y| 2 ≤y≤ 4, y∈ R}
c) The graph has been translated 3 units up. This is the vertical displacement. The graph has also been translated 30° to the right. This is called the phase shift.
Your Turn
a) Sketch the graph of the function y= cos (x+ 45°) − 2.
b) What are the domain and range of the function?
c) Use the language of transformations to compare your graph to the graph of y= cos x.
Graph y = a cos (θ− c) + d
a) Sketch the graph of the function y=−2 cos (θ+π) − 1over two cycles.
b) Use the language of transformations to compare your graph to the graph of y= cos θ. Indicate which parameter is related to each transformation.
Example 1
vertical displacement
the vertical translation •
of the graph of a periodic function
phase shift the horizontal •
translation of the graph of a periodic function
Solution
a) y
-1 1
0 π 2π 3π 4π
-2
-3
θ
b) Since a is −2, the graph has been reflected about the θ-axis and then stretched vertically by a factor of two. The d-value is −1, so the graph is translated 1 unit down. The sinusoidal axis is defined as y=−1. Finally, the c-value is -π. Therefore, the graph is translated π units to the left.
Your Turn
a) Sketch the graph of the function y= 2 sin
(
θ− _ π2
)
+ 2 over two cycles.b) Compare your graph to the graph of y= sin θ.
Graph y = a sin b(x - c) + d
Sketch the graph of the function y= 3 sin
(
2x- 2_ π3
)
+ 2 over two cycles. What are the vertical displacement, amplitude, period, phase shift, domain, and range for the function?Solution
First, rewrite the function in the standard form y=a sin b(x-c) +d. y= 3 sin 2
(
x- _ π3
)
+ 2Method 1: Graph Using Transformations
Step 1: Sketch the graph of y= sin x for one cycle. Apply the horizontal and vertical stretches to obtain the graph of y= 3 sin 2x.
Compared to the graph of y= sin x, the graph of y= 3 sin 2x is a horizontal stretch by a factor of 1 _
2 and a vertical stretch by a factor of 3. For the function y= 3 sin 2x, b= 2.
Period = 2_ π
|b| = 2_ π
2
=π
So, the period is π.
In this chapter, the parameters for horizontal and vertical translations are represented by c and d, respectively.
Did You Know?
Example 3
For the function y= 3 sin 2x, |a|= 3. So, the amplitude is 3.
y
x
π 2π 3π
-2 2 4
0
y= 3 sin 2x
y = sin x
Step 2: Apply the horizontal translation to obtain the graph of y= 3 sin 2
(
x- _ π3
)
.The phase shift is determined by the value of parameter c for a function in the standard form y=a sin b(x-c) +d.
Compared to the graph of y= 3 sin 2x, the graph of y= 3 sin 2
(
x- π_ 3)
is translated horizontally _ π3 units to the right. The phase shift is _ π
3 units to the right. y
x
π 2π 3π
-2 2 4
0
y = 3 sin 2
(
x -π3_)
y= 3 sin 2xStep 3: Apply the vertical translation to obtain the graph of y= 3 sin 2
(
x- _ π3
)
+ 2.The vertical displacement is determined by the value of parameter d for a function in the standard form y=a sin b(x-c) +d.
Compared to the graph of y= 3 sin 2
(
x- _ π3
)
, the graph of y= 3 sin 2(
x- _ π3
)
+ 2 is translated up 2 units. The vertical displacement is 2 units up.y
x
π 2π 3π 4π
-2 2 4 6
0
y= 3 sin 2
(
x -π_3)
+ 2y = 3 sin 2
(
x -π3_)
Would it matter if the order of the transformations were changed? Try a different order for the transformations.Compared to the graph of y= sin x, the graph of y= 3 sin 2
(
x- _ π 3)
+ 2 is horizontally stretched by a factor of• _ 1
2 vertically stretched by a factor of 3 •
horizontally translated
• _ π
3 units to the right vertically translated 2 units up
•
The vertical displacement is 2 units up. The amplitude is 3.
The phase shift is _ π
3 units to the right. The domain is {x|x∈ R}.
The range is {y|-1 ≤y≤ 5, y∈ R}. Method 2: Graph Using Key Points
You can identify five key points to graph one cycle of the sine function. The first, third, and fifth points indicate the start, the middle, and the end of the cycle. The second and fourth points indicate the maximum and minimum points.
Comparing y= 3 sin 2
(
x- π_3
)
+ 2 to y=a sin b(x-c) +d gives a= 3, b= 2, c= _ π3 , and d= 2. The amplitude is |a|, or 3.
The period is 2_ π
|b| , or π.
The vertical displacement is d, or 2. Therefore, the equation of the sinusoidal axis or mid-line is y= 2.
You can use the amplitude and vertical displacement to determine the maximum and minimum values.
The maximum value is d+|a| =2+3
= 5
The minimum value is d-|a| =2-3
=-1
Determine the values of x for the start and end of one cycle from the function y=a sin b(x-c) +d by solving the compound inequality 0 ≤b(x-c) ≤ 2π.
0 ≤2
(
x- _ π 3)
≤ 2π0 ≤x- _ π 3 ≤π
_ π
3 ≤x≤ 4
π
_
3
How does this inequality relate to the period of the function?
Divide the interval _ π
3 ≤x≤ 4
π
_
3 into four equal segments. By doing this, you can locate five key values of x along the sinusoidal axis.
_ π
3 , 7
π
_
12 , 5
π
_
6 , 13
π
_
12 , 4
π
_
3 y
x
π 2π 3π
2 4 6
0
y = 3 sin 2
(
x -π_3)
+ 2 Use the above information to sketchone cycle of the graph, and then a second cycle.
For the graph of the function y= 3 sin 2
(
x- _ π 3)
+ 2, the vertical displacement is 2 units up•
the amplitude is 3 •
the phase shift is
• _ π
3 units to the right the domain is {
• x|x∈ R}
the range is {
• y|-1 ≤y≤ 5, y∈ R}
Your Turn
Sketch the graph of the function y= 2 cos 4(x+π) - 1 over two cycles. What are the vertical displacement, amplitude, period, phase shift, domain, and range for the function?
Determine an Equation From a Graph
The graph shows the function y
x
π 2π
2 4
0 π_
3 2π__3 4π__3 5π__3 -π3_
y=f(x).
a) Write the equation of the function in the form
y=a sin b(x-c) +d, a> 0.
b) Write the equation of the function in the form
y=a cos b(x-c) +d, a> 0.
c) Use technology to verify your solutions.
Solution
a) Determine the values of the y
x
π 2π
2 4
0 π_
3 2π__3 4π__3 5π__3 -π3_
d= 2
a= 2
parameters a, b, c, and d.
Locate the sinusoidal axis or mid-line. Its position determines the value of d. Thus, d= 2.
Note the five key points and how you can use them to sketch one cycle of the graph of the function.
Use the sinusoidal axis from the graph or use the formula to determine the amplitude.
Amplitude = maximum value _______ - minimum value 2
a= 4__ -0 2 a= 2 The amplitude is 2.
Determine the period and the value of b.
Method 1: Count the Number of Cycles in 2π
Determine the number of cycles in a distance of 2π.
In this function, there are three cycles. Therefore, the value of b is 3 and the period is 2_ π
3 .
y
x
π 2π
2 4
-2 0 π_
3 2__3π 4__3π 5__3π
-π3_
Period
First
Cycle SecondCycle ThirdCycle
Method 2: Determine the Period First
Locate the start and end of one cycle of the sine curve.
Recall that one cycle of y = sin x starts at (0, 0). How is that point transformed? How could this information help you determine the start for one cycle of this sine curve?
The start of the first cycle of the sine curve that is closest to the y-axis is at x= _ π
6 and the end is at x= 5_ π
6 .
The period is 5_ π 6 -
π
_
6 , or 2
π
_
3 . Solve the equation for b.
Period = 2_ π
|b|
_ 2π
3 = 2
π
_
|b|
b= 3
Determine the phase shift, c.
Locate the start of the first cycle of the sine curve to the right of the y-axis. Thus, c= _ π
6 .
Substitute the values of the parameters a= 2, b= 3, c= _ π 6 , and d= 2 into the equation y=a sin b(x-c) +d.
The equation of the function in the form y=a sin b(x-c) +d is y= 2 sin 3
(
x- _ π6
)
+ 2.How can you use the maximum and minimum
values of the graph to find the value of d?
Choose b to be positive.
b) To write an equation in the form y=a cos b(x-c) +d, determine the values of the parameters a, b, c, and d using steps similar to what you did for the sine function in part a).
a = 2 b= 3 c= π_
3 d= 2
y
x
π 2π
2 4
-2 0 π_
3 2__3π 4__3π 5__3π
-π3_
Period
The equation of the function in the form y=a cos b(x-c) +d is y= 2 cos 3
(
x- _ π3
)
+ 2.c) Enter the functions on a graphing calculator. Compare the graphs to the original and to each other.
The graphs confirm that the equations for the function are correct.
Your Turn
The graph shows the function y=f(x). y
x π 2
-2 0 π_
3 2π__3
a) Write the equation of the function in the form y =a sin b(x-c) +d, a> 0.
b) Write the equation of the function in the form y=a cos b(x-c) +d, a> 0.
c) Use technology to verify your solutions.
Why is c = π_3 ? Are there other possible values for c?
How do the two equations compare?
Could other equations define the
Interpret Graphs of Sinusoidal Functions
Prince Rupert, British Columbia, has the deepest natural harbour in North America. The depth, d, in metres, of the berths for the ships can be approximated by the equation d(t) = 8 cos π_
6 t+ 12, where t is the time, in hours, after the first high tide.
a) Graph the function for two cycles.
b) What is the period of the tide?
c) An ocean liner requires a minimum of 13 m of water to dock safely. From the graph, determine the number of hours per cycle the ocean liner can safely dock.
d) If the minimum depth of the berth occurs at 6 h, determine the depth of the water. At what other times is the water level at a minimum? Explain your solution.
Solution
a) d
t
6 9 12
3 15 18 21 24
4 8 12 16 20
0
Depth (m)
Time (h)
Depth of Berths for Prince Rupert Harbour
b) Use b= _ π
6 to determine the period. Period = 2_ π
|b|
Period = 2
_
π|
_ π6
|
Period = 12
The period for the tides is 12 h.
Example 5
Why should you set the calculator to radian mode when graphing sinusoidal functions that represent real-world situations?
What does the period of 12 h represent?
c) To determine the number of hours an ocean liner can dock safely, draw the line y= 13 to represent the minimum depth of the berth. Determine the points of intersection of the graphs of y= 13 and d(t) = 8 cos _ π
6 t+ 12.
The points of intersection for the first cycle are approximately (2.76, 13) and (9.26, 13).
The depth is greater than 13 m from 0 h to approximately 2.76 h and from approximately 9.24 h to 12 h. The total time when the depth is greater than 13 m is 2.76 + 2.76, or 5.52 h, or about 5 h 30 min per cycle.
d) To determine the berth depth at 6 h, substitute the value of t= 6 into the equation.
d(t) = 8 cos _ π 6 t+ 12 d(6) = 8 cos _ π
6 (6) + 12 d(6) = 8 cos π+ 12 d(6) = 8(-1) + 12 d(6) = 4
The berth depth at 6 h is 4 m. Add 12 h (the period) to 6 h to determine the next time the berth depth is 4 m. Therefore, the berth depth of 4 m occurs again at 18 h.
Your Turn
The depth, d, in metres, of the water in the harbour at New Westminster, British Columbia, is approximated by the equation d(t) = 0.6 cos 2_ π
13 t+ 3.7, where t is the time, in hours, after the first high tide.
a) Graph the function for two cycles starting at t= 0.
b) What is the period of the tide?
c) If a boat requires a minimum of 3.5 m of water to launch safely, for how many hours per cycle can the boat safely launch?
d) What is the depth of the water at 7 h? At what other times is the water level at this depth? Explain your solution.
More precise answers can be obtained using technology.
You can use the graph to verify the solution.
Key Ideas
You can determine the amplitude, period, phase shift, and vertical displacement of sinusoidal functions when the equation of the function is given in the form y=a sin b(x-c) +d or y=a cos b(x-c) +d. For: y=a sin b(x-c) +d
y=a cos b(x-c) +d
3 y
x
π π
1 2
0 π_
2 π_
4 3π__4
2π __ |b|
5π__
4 3π__2 7π__4 -π_4
d
c
a
-1
Vertical stretch by a factor of |a|
• changes the amplitude to |a|
• reflected in the x-axis if a< 0 Horizontal stretch by a factor of 1 _
|b|
• changes the period to 360° _
|b| (in degrees) or 2 π
_
|b| (in radians)
• reflected in the y-axis if b< 0
Horizontal phase shift represented by c • to right if c> 0
• to left if c< 0
Vertical displacement represented by d • up if d> 0
• down if d< 0
d= maximum value _______ + minimum value 2
You can determine the equation of a sinusoidal function given its properties or its graph.
How does changing each parameter affect the graph of a function?
Check Your Understanding
Practise
1. Determine the phase shift and the vertical displacement with respect to y= sin x for each function. Sketch a graph of each function.
a) y = sin (x- 50°) + 3
b) y= sin (x+π)
c) y= sin
(
x+ 2_ π 3)
+ 5d) y = 2 sin (x+ 50°) - 10
e) y =-3 sin (6x+ 30°) - 3
f) y= 3 sin 1 _ 2
(
x-π
_
4
)
- 102. Determine the phase shift and the vertical displacement with respect to y= cos x for each function. Sketch a graph of each function.
a) y = cos (x- 30°) + 12
b) y= cos
(
x- π_ 3)
c) y= cos
(
x+ 5_ π 6)
+ 16d) y = 4 cos (x+ 15°) + 3
e) y= 4 cos (x-π) + 4
f) y= 3 cos
(
2x- _ π 6)
+ 73. a) Determine the range of each function.
i) y= 3 cos
(
x- _ π 2)
+ 5ii) y=-2 sin (x+π) - 3
iii)y= 1.5 sin x+ 4
iv)y= 2 _
3 cos (x+ 50°) + 3 _ 4
b) Describe how to determine the range when given a function of the form y=a cos b(x-c) +d or
y=a sin b(x-c) +d.
4. Match each function with its description in the table.
a) y=-2 cos 2(x+ 4) - 1
b) y= 2 sin 2(x- 4) - 1
c) y= 2 sin (2x- 4) - 1
d) y= 3 sin (3x- 9) - 1
e) y= 3 sin (3x+π) - 1
Amplitude Period Phase Shift Vertical Displacement A 3 2π _
3 3 right 1 down
B 2 π 2 right 1 down
C 2 π 4 right 1 down
D 2 π 4 left 1 down
E 3 2π _ 3 π _
3 left 1 down
5. Match each function with its graph.
a) y= sin
(
x- _ π 4)
b) y= sin
(
x+ _ π 4)
c) y= sin x- 1
d) y= sin x+ 1
A y
x
π 2π
2
0 π_
2 3__2π
-π2_
B y
x
π 2π
2
-2 0 π_
2 3__2π
-π2_
C y
x
π 2π -2
0 π_
2 3__2π
-π2_
D
π_ 2
y
x
π 2π
2
Apply
6. Write the equation of the sine function in the form y=a sin b(x-c) +d given its characteristics.
a) amplitude 4, period π, phase shift _ π 2 to the right, vertical displacement 6 units down
b) amplitude 0.5, period 4π, phase shift _ π
6 to the left, vertical displacement 1 unit up
c) amplitude 3 _
4 , period 720°, no phase shift, vertical displacement 5 units down
7. The graph of y= cos x is transformed as described. Determine the values of the parameters a, b, c, and d for the transformed function. Write the equation for the transformed function in the form y=a cos b(x-c) +d.
a) vertical stretch by a factor of 3 about the x-axis, horizontal stretch by a factor of 2 about the y-axis, translated 2 units to the left and 3 units up
b) vertical stretch by a factor of 1 _ 2 about the x-axis, horizontal stretch by a factor of 1 _
4 about the y-axis, translated 3 units to the right and 5 units down
c) vertical stretch by a factor of 3 _ 2 about the x-axis, horizontal stretch by a factor of 3 about the y-axis, reflected in the x-axis, translated _ π4 units to the right and 1 unit down
8. When white light shines through a prism, the white light is broken into the colours of the visible light spectrum. Each colour corresponds to a different wavelength of the electromagnetic spectrum. Arrange the colours, in order from greatest to smallest period.
Blue
Red
Green
Indigo
Violet
Orange
Yellow
9. The piston engine is the most commonly used engine in the world. The height of the piston over time can be modelled by a sine curve. Given the equation for a sine curve, y=a sin b(x -c) +d, which parameter(s) would be affected as the piston moves faster?
3π
__ 2 5__π
4 y
x
π
2
-2
0 π_
2
π_
4 3__4π
Height (cm)
Time (s)
10. Victor and Stewart determined the phase shift for the function f(x) = 4 sin (2x- 6) + 12. Victor said that the phase shift was 6 units to the right, while Stewart claimed it was 3 units to the right.
a) Which student was correct? Explain your reasoning.
b) Graph the function to verify your answer from part a).
11. A family of sinusoidal graphs with equations of the form
y=a sin b(x-c) +d is created by changing only the vertical displacement of the function. If the range of the original function is {y|-3 ≤y≤ 3, y∈ R},
determine the range of the function with each given value of d.
a) d= 2
b) d=-3
c) d=-10
d) d= 8
12. Sketch the graph of the curve that results after applying each transformation to the graph of the function f(x) = sin x.
a) f
(
x- _ π 3)
b) f
(
x+ _ π 4)
c) f(x) + 3
d) f(x) - 4
13. The range of a trigonometric function in the form y=a sin b(x-c) +d is {y|-13 ≤y≤ 5, y∈ R}. State the values of a and d.
14. For each graph of a sinusoidal function, state
i) the amplitude
ii) the period
iii) the phase shift
iv) the vertical displacement
v) the domain and range
vi) the maximum value of y and the values of x for which it occurs over the interval 0 ≤x≤ 2π
vii) the minimum value of y and the values of x for which it occurs over the interval 0 ≤x≤ 2π
a) a sine function 3y
x
π
-π 2π
-1
-2
-3 1 2
0 π_ 2
-π_2 3__2π -3__2π
b) a cosine function y
x
π
-π 2π
-1
-2
-3
-4 0 π_
2
-π_2 3__2π -3__2π
c) a sine function 3y
x
π 2π 3π -1
1 2
0 π_ 2
15. Determine an equation in the form y=a sin b(x-c) +d for each graph.
a) y
x
π -π
-2 2
0
b)
4 y
x
π -π
-2 2
0
c)
4 y
x 2
0 π_
2
-π_2
16. For each graph, write an equation in the form y=a cos b(x-c) +d.
a)
-π
y
x
π -2
2
0 π_ 2
-π2_ 3__2π
b)
-π
y
x
π -2
2
0 π_ 2
-π2_ 3π__2
c) y
x 2π 3π 4π 5π π
2
0
17. a) Graph the function f(x) = cos
(
x- _ π 2)
.b) Consider the graph. Write an equation of the function in the form y=a sin b(x-c) +d.
c) What conclusions can you make about the relationship between the two equations of the function?
18. Given the graph of the function f(x) = sin x, what transformation is required so that the function g(x) = cos x describes the graph of the image function?
19. For each start and end of one cycle of a cosine function in the form y= 3 cos b(x-c),
i) state the phase shift, period, and x-intercepts
ii) state the coordinates of the minimum and maximum values
a) 30° ≤x≤ 390°
b) _ π
4 ≤x≤ 5
π
_
4
20. The Wave is a spectacular sandstone formation on the slopes of the Coyote Buttes of the Paria Canyon in Northern Arizona. The Wave is made from 190 million-year-old sand dunes that have turned to red rock. Assume that a cycle of the Wave may be approximated using a cosine curve. The maximum height above sea level is 5100 ft and the minimum height is 5000 ft. The beginning of the cycle is at the 1.75 mile mark of the canyon and the end of this cycle is at the 2.75 mile mark. Write an equation that approximates the pattern of the Wave.
21. Compare the graphs of the functions y= 3 sin _ π
3 (x- 2) - 1 and y= 3 cos _ π
3
(
x- 7 _ 2)
- 1. Are the graphs equivalent? Support your answer graphically.22. Noise-cancelling headphones are designed to give you maximum listening pleasure by cancelling ambient noise and actively creating their own sound waves. These waves mimic the incoming noise in every way, except that they are out of sync with the intruding noise by 180°.
sound waves created by headphones
noise created by outside source
combining the two sound waves results in silence
Suppose that the amplitude and period for the sine waves created by the outside noise are 4 and _ π
2 , respectively. Determine the equation of the sound waves the headphones produce to effectively cancel the ambient noise.
23. The overhang of the roof of a house is designed to shade the windows for cooling in the summer and allow the Sun’s rays to enter the house for heating in the winter. The Sun’s angle of elevation, A, in degrees, at noon in Estevan, Saskatchewan, can be modelled by the formula A=-23.5 sin 360 _
365 (x+ 102) + 41, where x is the number of days elapsed beginning with January 1.
a) Use technology to sketch the graph showing the changes in the Sun’s angle of elevation throughout the year.
b) Determine the Sun’s angle of elevation at noon on February 12.
c) On what date is the angle of elevation the greatest in Estevan?
24. After exercising for 5 min, a person has a respiratory cycle for which the rate of air flow, r, in litres per second, in the lungs is approximated by r= 1.75 sin _ π
2 t, where t is the time, in seconds.
a) Determine the time for one full respiratory cycle.
b) Determine the number of cycles per minute.
c) Sketch the graph of the rate of air flow function.
d) Determine the rate of air flow at a time of 30 s. Interpret this answer in the context of the respiratory cycle.
e) Determine the rate of air flow at a time of 7.5 s. Interpret this answer in the context of the respiratory cycle.
Extend
25. The frequency of a wave is the number of cycles that occur in 1 s. Adding two sinusoidal functions with similar, but unequal, frequencies results in a function that pulsates, or exhibits beats. Piano tuners often use this phenomenon to help them tune a piano.
a) Graph the function y = cos x+ cos 0.9x.
b) Determine the amplitude and the period of the resulting wave.
26. a) Copy each equation. Fill in the missing values to make the equation true.
i) 4 sin (x- 30°) = 4 cos (x- )
ii) 2 sin
(
x- _ π4
)
= 2 cos (x- )iii)-3 cos
(
x- _ π2
)
= 3 sin (x+ )iv) cos (-2x+ 6π) = sin 2(x+ )
b) Choose one of the equations in part a) and explain how you got your answer.
